pacman::p_load(sf, tidyverse, funModeling,
blorr, corrplot, ggpubr,
sf, spdep, GWmodel,
tmap, skimr, caret)In-class Exercise 5
Objective
In this in-class exercise, we wish to build a logistic regression model for the water point status (functional or non-functional) at Osun state, Nigeria.
Getting started
The code chunk below installs and loads the following R packages:
sf
tidyverse
funModeling
blorr - for logistic regression
corrplot
ggpubr
sf
spdep
GWmodel
tmap
skimr - to do EDA
caret - for error matrix and comparison (of models?)
Importing the Analytical Data
The code chunk below brings the datasets into R.
osun <- read_rds("rds/Osun.rds")wp <- read_rds("rds/Osun_wp_sf.rds")osun contains the ADM2 polygon boundaries for Osun state of Nigeria, and wp is the water point data in Osun state of Nigeria.
The rds files have been pre-processed and wrangled (e.g. cleaning up of variables and variable names).
Next, we check the status field of the wp sf data frame object. Note that the data type for this field is logi, i.e. it only takes the values of TRUE or FALSE. This was recoded from the field status_clean of the water point data set, where
observations without the status information (
NaN) are filtered away,all remaining values that indicates that the water point is functional are recoded as
T, andthe rest are recoded as
F.
wp %>%
freq(input = "status")
status frequency percentage cumulative_perc
1 TRUE 2642 55.5 55.5
2 FALSE 2118 44.5 100.0We see that there are 2,642 TRUE values and 2,118 FALSE values.
Bear in mind that linear and logistic regression models do not like missing values - the entire record across all variables would be removed if one or more of the variables have missing values. Hence, it is important to check upfront for missing values and exclude variables which have a significant proportion of missing values (e.g. 20%).
Next, we plot the distribution of the water points by status using tmap, as shown in the code chunk below.
tmap_mode("view")
tm_shape(osun) +
tmap_options(check.and.fix = TRUE) +
tm_polygons(alpha = 0.4) +
tm_shape(wp) +
tm_dots(col = "status",
alpha = 0.6) +
tm_view(set.zoom.limits = c(9, 12))We set tmap_mode() back to “plot” option after plotting.
tmap_mode("plot")Exploratory Data Analysis (EDA)
We look at the summary statistics of the water point dataset with skimr for preliminary variable selection, in the following code chunk.
wp %>%
skim()| Name | Piped data |
| Number of rows | 4760 |
| Number of columns | 75 |
| _______________________ | |
| Column type frequency: | |
| character | 47 |
| logical | 5 |
| numeric | 23 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| source | 0 | 1.00 | 5 | 44 | 0 | 2 | 0 |
| report_date | 0 | 1.00 | 22 | 22 | 0 | 42 | 0 |
| status_id | 0 | 1.00 | 2 | 7 | 0 | 3 | 0 |
| water_source_clean | 0 | 1.00 | 8 | 22 | 0 | 3 | 0 |
| water_source_category | 0 | 1.00 | 4 | 6 | 0 | 2 | 0 |
| water_tech_clean | 24 | 0.99 | 9 | 23 | 0 | 3 | 0 |
| water_tech_category | 24 | 0.99 | 9 | 15 | 0 | 2 | 0 |
| facility_type | 0 | 1.00 | 8 | 8 | 0 | 1 | 0 |
| clean_country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| clean_adm1 | 0 | 1.00 | 3 | 5 | 0 | 5 | 0 |
| clean_adm2 | 0 | 1.00 | 3 | 14 | 0 | 35 | 0 |
| clean_adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| clean_adm4 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| installer | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management_clean | 1573 | 0.67 | 5 | 37 | 0 | 7 | 0 |
| status_clean | 0 | 1.00 | 9 | 32 | 0 | 7 | 0 |
| pay | 0 | 1.00 | 2 | 39 | 0 | 7 | 0 |
| fecal_coliform_presence | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| subjective_quality | 0 | 1.00 | 18 | 20 | 0 | 4 | 0 |
| activity_id | 4757 | 0.00 | 36 | 36 | 0 | 3 | 0 |
| scheme_id | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| wpdx_id | 0 | 1.00 | 12 | 12 | 0 | 4760 | 0 |
| notes | 0 | 1.00 | 2 | 96 | 0 | 3502 | 0 |
| orig_lnk | 4757 | 0.00 | 84 | 84 | 0 | 1 | 0 |
| photo_lnk | 41 | 0.99 | 84 | 84 | 0 | 4719 | 0 |
| country_id | 0 | 1.00 | 2 | 2 | 0 | 1 | 0 |
| data_lnk | 0 | 1.00 | 79 | 96 | 0 | 2 | 0 |
| water_point_history | 0 | 1.00 | 142 | 834 | 0 | 4750 | 0 |
| clean_country_id | 0 | 1.00 | 3 | 3 | 0 | 1 | 0 |
| country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| water_source | 0 | 1.00 | 8 | 30 | 0 | 4 | 0 |
| water_tech | 0 | 1.00 | 5 | 37 | 0 | 20 | 0 |
| adm2 | 0 | 1.00 | 3 | 14 | 0 | 33 | 0 |
| adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management | 1573 | 0.67 | 5 | 47 | 0 | 7 | 0 |
| adm1 | 0 | 1.00 | 4 | 5 | 0 | 4 | 0 |
| New Georeferenced Column | 0 | 1.00 | 16 | 35 | 0 | 4760 | 0 |
| lat_lon_deg | 0 | 1.00 | 13 | 32 | 0 | 4760 | 0 |
| public_data_source | 0 | 1.00 | 84 | 102 | 0 | 2 | 0 |
| converted | 0 | 1.00 | 53 | 53 | 0 | 1 | 0 |
| created_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| updated_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| Geometry | 0 | 1.00 | 33 | 37 | 0 | 4760 | 0 |
| ADM2_EN | 0 | 1.00 | 3 | 14 | 0 | 30 | 0 |
| ADM2_PCODE | 0 | 1.00 | 8 | 8 | 0 | 30 | 0 |
| ADM1_EN | 0 | 1.00 | 4 | 4 | 0 | 1 | 0 |
| ADM1_PCODE | 0 | 1.00 | 5 | 5 | 0 | 1 | 0 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| rehab_year | 4760 | 0 | NaN | : |
| rehabilitator | 4760 | 0 | NaN | : |
| is_urban | 0 | 1 | 0.39 | FAL: 2884, TRU: 1876 |
| latest_record | 0 | 1 | 1.00 | TRU: 4760 |
| status | 0 | 1 | 0.56 | TRU: 2642, FAL: 2118 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| row_id | 0 | 1.00 | 68550.48 | 10216.94 | 49601.00 | 66874.75 | 68244.50 | 69562.25 | 471319.00 | ▇▁▁▁▁ |
| lat_deg | 0 | 1.00 | 7.68 | 0.22 | 7.06 | 7.51 | 7.71 | 7.88 | 8.06 | ▁▂▇▇▇ |
| lon_deg | 0 | 1.00 | 4.54 | 0.21 | 4.08 | 4.36 | 4.56 | 4.71 | 5.06 | ▃▆▇▇▂ |
| install_year | 1144 | 0.76 | 2008.63 | 6.04 | 1917.00 | 2006.00 | 2010.00 | 2013.00 | 2015.00 | ▁▁▁▁▇ |
| fecal_coliform_value | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| distance_to_primary_road | 0 | 1.00 | 5021.53 | 5648.34 | 0.01 | 719.36 | 2972.78 | 7314.73 | 26909.86 | ▇▂▁▁▁ |
| distance_to_secondary_road | 0 | 1.00 | 3750.47 | 3938.63 | 0.15 | 460.90 | 2554.25 | 5791.94 | 19559.48 | ▇▃▁▁▁ |
| distance_to_tertiary_road | 0 | 1.00 | 1259.28 | 1680.04 | 0.02 | 121.25 | 521.77 | 1834.42 | 10966.27 | ▇▂▁▁▁ |
| distance_to_city | 0 | 1.00 | 16663.99 | 10960.82 | 53.05 | 7930.75 | 15030.41 | 24255.75 | 47934.34 | ▇▇▆▃▁ |
| distance_to_town | 0 | 1.00 | 16726.59 | 12452.65 | 30.00 | 6876.92 | 12204.53 | 27739.46 | 44020.64 | ▇▅▃▃▂ |
| rehab_priority | 2654 | 0.44 | 489.33 | 1658.81 | 0.00 | 7.00 | 91.50 | 376.25 | 29697.00 | ▇▁▁▁▁ |
| water_point_population | 4 | 1.00 | 513.58 | 1458.92 | 0.00 | 14.00 | 119.00 | 433.25 | 29697.00 | ▇▁▁▁▁ |
| local_population_1km | 4 | 1.00 | 2727.16 | 4189.46 | 0.00 | 176.00 | 1032.00 | 3717.00 | 36118.00 | ▇▁▁▁▁ |
| crucialness_score | 798 | 0.83 | 0.26 | 0.28 | 0.00 | 0.07 | 0.15 | 0.35 | 1.00 | ▇▃▁▁▁ |
| pressure_score | 798 | 0.83 | 1.46 | 4.16 | 0.00 | 0.12 | 0.41 | 1.24 | 93.69 | ▇▁▁▁▁ |
| usage_capacity | 0 | 1.00 | 560.74 | 338.46 | 300.00 | 300.00 | 300.00 | 1000.00 | 1000.00 | ▇▁▁▁▅ |
| days_since_report | 0 | 1.00 | 2692.69 | 41.92 | 1483.00 | 2688.00 | 2693.00 | 2700.00 | 4645.00 | ▁▇▁▁▁ |
| staleness_score | 0 | 1.00 | 42.80 | 0.58 | 23.13 | 42.70 | 42.79 | 42.86 | 62.66 | ▁▁▇▁▁ |
| location_id | 0 | 1.00 | 235865.49 | 6657.60 | 23741.00 | 230638.75 | 236199.50 | 240061.25 | 267454.00 | ▁▁▁▁▇ |
| cluster_size | 0 | 1.00 | 1.05 | 0.25 | 1.00 | 1.00 | 1.00 | 1.00 | 4.00 | ▇▁▁▁▁ |
| lat_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| lon_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| count | 0 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | ▁▁▇▁▁ |
For example, we note that 1/4 of install_year are missing values. Hence, despite the variable being useful, we need to drop this variable from the logistic regression model building.
Based on the EDA, we will use the following variables for our model building, in the code chunk below.
wp_clean <- wp %>%
filter_at(vars(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean),
all_vars(!is.na(.))) %>%
mutate(usage_capacity = as.factor(usage_capacity))What we have done above are to:
exclude missing values (filtering for
all_vars(!is.na(.))); andrecode usage_capacity as factor (it only has 3 classes) instead of numerical data type. This is because the calibration of logit function will be different.
We also check the wp_clean object in the Environment panel. We see that the number of observations dropped by 4 from 4,760 to 4,756, which signifies that the 4 missing values from the water_point_population and local_population_1km fields are successfully removed. We also check that the status field has been correctly recoded to Factor w/ 2 levels "300", "1000".
Correlation Analysis
We first extract the desired variables from wp_clean into a new object osun_wp, and remove the geometry column by setting st_set_geometry() to NULL so that we can do a correlation matrix plot.
osun_wp <- wp_clean %>%
select(c(7, 35:39, 42:43, 46:47, 57)) %>%
st_set_geometry(NULL)Next, we plot the correlation matrix for all the numerical data fields.
cluster_vars.cor = cor(
osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
lower = "ellipse",
upper = "number",
tl.pos = "lt",
diag = "l",
tl.col = "black")
We see that none of the variables are highly correlated with any other variable (r no more than 0.85). Hence, we will keep all the variables for logistic regression model building in the next section.
Building a Logistic Regression Model
In the code chunk below, glm() of R stats is used to calibrate a logistic regression for the water point status.
model <- glm(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_clean,
family = binomial(link = "logit"))Instead of using a typical R report, we use blr_regress() of blorr to generate the model report in scientific literature reporting format.
blr_regress(model) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4744 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3887 0.1124 3.4588 5e-04
distance_to_primary_road 1 0.0000 0.0000 -0.7153 0.4744
distance_to_secondary_road 1 0.0000 0.0000 -0.5530 0.5802
distance_to_tertiary_road 1 1e-04 0.0000 4.6708 0.0000
distance_to_city 1 0.0000 0.0000 -4.7574 0.0000
distance_to_town 1 0.0000 0.0000 -4.9170 0.0000
is_urbanTRUE 1 -0.2971 0.0819 -3.6294 3e-04
usage_capacity1000 1 -0.6230 0.0697 -8.9366 0.0000
water_source_cleanProtected Shallow Well 1 0.5040 0.0857 5.8783 0.0000
water_source_cleanProtected Spring 1 1.2882 0.4388 2.9359 0.0033
water_point_population 1 -5e-04 0.0000 -11.3686 0.0000
local_population_1km 1 3e-04 0.0000 19.2953 0.0000
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7347 Somers' D 0.4693
% Discordant 0.2653 Gamma 0.4693
% Tied 0.0000 Tau-a 0.2318
Pairs 5585188 c 0.7347
---------------------------------------------------------------At 95% confidence level, variables with p-value less than 0.05 are statistically significant. These are all independent variables except distance_to_primary_road and distance_to_secondary_road.
For interpretation of logistic regression report:
Categorical variables: A positive value implies an above average correlation and a negative value implies a below average correlation, while the magnitude of the coefficient does not matter for categorical variables;
Continuous variables: a positive value implies a direct correlation and a negative value implies an inverse correlation, while the magnitude of the value gives the strength of the correlation.
We generate the confusion matrix for the model using blr_confusion_matrix() of blorr.
blr_confusion_matrix(model, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1301 738
1 813 1904
Accuracy : 0.6739
No Information Rate : 0.4445
Kappa : 0.3373
McNemars's Test P-Value : 0.0602
Sensitivity : 0.7207
Specificity : 0.6154
Pos Pred Value : 0.7008
Neg Pred Value : 0.6381
Prevalence : 0.5555
Detection Rate : 0.4003
Detection Prevalence : 0.5713
Balanced Accuracy : 0.6680
Precision : 0.7008
Recall : 0.7207
'Positive' Class : 1The validity of a cut-off (here we use 0.5) is measured using sensitivity, specificity and accuracy.
Sensitivity: The % of correctly classified positive events out of all predicted positive events = TP / (TP + FN).
Specificity: The % of correctly classified negative events out of all predicted negative events = TN / (TN + FP).
Accuracy: The % of correctly classified events out of all events = (TP + TN) / (TP + FP + TN + FN).
Note that TP = true positve, TN = true negative, FP = false positive and FN = false negative.
From the output, we see that the model gives us an accuracy of 0.6739, which is a good start as it is better than guessing (0.5).
The sensitivity and specificity are 0.7207 and 0.6154 respectively. This shows that the true positives (functional water points) are slightly higher than the true negative prediction rates (non-functional water points).
Building Geographically Weighted Logistic Regression (gwLR) Model
Converting from sf to sp Data Frame
Next, we need to convert the sf data frame into spatial point data frame for GWR model building. This is done using the code chunk below.
wp_sp <- wp_clean %>%
select(c(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
is_urban,
usage_capacity,
water_source_clean)) %>%
as_Spatial()
wp_spclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 11
names : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean
min values : 0, 0.014461356813335, 0.152195902540837, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 0, 1000, Borehole
max values : 1, 26909.8616132094, 19559.4793799085, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 1, 300, Protected Spring Note that we use the cleaned version of the water point sf data frame for consistency in the geometrics with our model building (4 water points with missing values excluded).
Building Fixed Bandwidth GWR Model
bw.fixed <- bw.ggwr(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_sp,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE, # for fixed bandwidth
longlat = FALSE) # input data have been converted to projected CRSModel Assessment
bw.fixed[1] 2599.672
We get the above output. We feed it into the bw argument in ggwr.basic() of GWmodel in the code chunk below.
gwlr.fixed <- ggwr.basic(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_sp,
bw = 2599.672,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405 We look at the results below. Similar to when we build multiple linear regression model, the report has 2 sections - generalised regression (global model) results and geographically weighted (GW) regression results. Note that the global model does not have AICc result, so AIC should be used to compare the 2 models.
gwlr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 16:40:22
Call:
ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road +
distance_to_tertiary_road + distance_to_city + distance_to_town +
is_urban + usage_capacity + water_source_clean + water_point_population +
local_population_1km, data = wp_sp, bw = 2599.672, family = "binomial",
kernel = "gaussian", adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-124.555 -1.755 1.072 1.742 34.333
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.887e-01 1.124e-01 3.459 0.000543
distance_to_primary_road -4.642e-06 6.490e-06 -0.715 0.474422
distance_to_secondary_road -5.143e-06 9.299e-06 -0.553 0.580230
distance_to_tertiary_road 9.683e-05 2.073e-05 4.671 3.00e-06
distance_to_city -1.686e-05 3.544e-06 -4.757 1.96e-06
distance_to_town -1.480e-05 3.009e-06 -4.917 8.79e-07
is_urbanTRUE -2.971e-01 8.185e-02 -3.629 0.000284
usage_capacity1000 -6.230e-01 6.972e-02 -8.937 < 2e-16
water_source_cleanProtected Shallow Well 5.040e-01 8.574e-02 5.878 4.14e-09
water_source_cleanProtected Spring 1.288e+00 4.388e-01 2.936 0.003325
water_point_population -5.097e-04 4.484e-05 -11.369 < 2e-16
local_population_1km 3.451e-04 1.788e-05 19.295 < 2e-16
Intercept ***
distance_to_primary_road
distance_to_secondary_road
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
is_urbanTRUE ***
usage_capacity1000 ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
water_point_population ***
local_population_1km ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.0 on 4744 degrees of freedom
AIC: 5712
Number of Fisher Scoring iterations: 5
AICc: 5712.099
Pseudo R-square value: 0.1295351
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2599.672
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -8.7229e+02 -4.9955e+00 1.7600e+00
distance_to_primary_road -1.9389e-02 -4.8031e-04 2.9618e-05
distance_to_secondary_road -1.5921e-02 -3.7551e-04 1.2317e-04
distance_to_tertiary_road -1.5618e-02 -4.2368e-04 7.6179e-05
distance_to_city -1.8416e-02 -5.6217e-04 -1.2726e-04
distance_to_town -2.2411e-02 -5.7283e-04 -1.5155e-04
is_urbanTRUE -1.9790e+02 -4.2908e+00 -1.6864e+00
usage_capacity1000 -2.0772e+01 -9.7231e-01 -4.1592e-01
water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01 5.3340e-01
water_source_cleanProtected.Spring -5.2235e+02 -5.5977e+00 2.5441e+00
water_point_population -5.2208e-02 -2.2767e-03 -9.8875e-04
local_population_1km -1.2698e-01 4.9952e-04 1.0638e-03
3rd Qu. Max.
Intercept 1.2763e+01 1073.2156
distance_to_primary_road 4.8443e-04 0.0142
distance_to_secondary_road 6.0692e-04 0.0258
distance_to_tertiary_road 6.6815e-04 0.0128
distance_to_city 2.3718e-04 0.0150
distance_to_town 1.9271e-04 0.0224
is_urbanTRUE 1.2841e+00 744.3099
usage_capacity1000 3.0322e-01 5.9281
water_source_cleanProtected.Shallow.Well 1.7849e+00 67.6343
water_source_cleanProtected.Spring 6.7663e+00 317.4133
water_point_population 5.0102e-04 0.1309
local_population_1km 1.8157e-03 0.0392
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2795.084
AIC : 4414.606
AICc : 4747.423
Pseudo R-square value: 0.5722559
***********************************************************************
Program stops at: 2022-12-17 16:40:54 Comparing the AIC values of the 2 models, we see that it is lower for the GW regression model at 4,414.606 then for the global regression model at 5,712.
Converting SDF into sf Data Frame
To assess the performance of the gwLR, we will first convert the SDF object in as data frame by using te code chunk below.
gwr.fixed <- as.data.frame(gwlr.fixed$SDF)Next, we will label yhat values (probability values for functional water points) greater or equal to 0.5 as 1 and otherwise 0. The result of the logic comparison operation will be saved into a field called most.
gwr.fixed <- gwr.fixed %>%
mutate(most = ifelse(
gwr.fixed$yhat >= 0.5, T, F))Confusion Matrix
Next, we use confusionMatrix() of caret to display the confusion matrix of the GW model using fixed bandwidth method.
gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data = gwr.fixed$most, # predicted outcome
reference = gwr.fixed$y, # reference y
positive = "TRUE") # setting positive class to "TRUE"
CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1824 263
TRUE 290 2379
Accuracy : 0.8837
95% CI : (0.8743, 0.8927)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7642
Mcnemar's Test P-Value : 0.2689
Sensitivity : 0.9005
Specificity : 0.8628
Pos Pred Value : 0.8913
Neg Pred Value : 0.8740
Prevalence : 0.5555
Detection Rate : 0.5002
Detection Prevalence : 0.5612
Balanced Accuracy : 0.8816
'Positive' Class : TRUE
We see that the accuracy (0.8837 vs 0.6739), sensitivity (0.9005 vs 0.7207) and specificity (0.8628 vs 0.6154) values have all improved from the non-gwLR global model. By using the gwLR model, we can explain the functional and non-functional water points better now which allows better management of water points through localised strategies (e.g. look at the local neighbourhood regions within Osun state).
Visualising gwLR
Before we visualise the results of the gwLR model, we clean up the data set for plotting by selecting the relevant data fields (mainly the status column which is the dependent or predicted variable) into a new sf data frame object wp_sf_select in the code chunk below.
wp_sf_select <- wp_clean %>%
select(c(ADM2_EN, ADM2_PCODE,
ADM1_EN, ADM1_PCODE,
status))We then combine it with gwr.fixed which has the predicted values of the water point status, in the form of probabilities between 0 and 1.
gwr_sf.fixed <- cbind(wp_sf_select, gwr.fixed)The code chunk below is used to create an interactive point symbol map.
tmap_mode("view")
actual <- tm_shape(osun) +
tmap_options(check.and.fix = TRUE) +
tm_polygons(alpha = 0.4) +
tm_shape(wp) +
tm_dots(col = "status",
alpha = 0.6,
palette = "YlOrRd") +
tm_view(set.zoom.limits = c(9, 12))
prob_T <- tm_shape(osun) +
tm_polygons(alpha = 0.4) +
tm_shape(gwr_sf.fixed) +
tm_dots(col = "yhat",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(9, 12))
tmap_arrange(actual, prob_T,
asp = 1, ncol = 2, sync = TRUE)We see that the predictions are largely aligned with the actual status of the water points, in line with the 88% accuracy rate.
Employing Only Statistically Significant Variables in Global and gwLR Models
As we earlier saw that 2 of the 10 variables, distance_to_primary_road and distance_to_secondary_road, are not statistically significant (p-values > 0.05), we should build logistic regression models without these 2 variables.
Hence, we repeat the relevant steps above to replicate the model building, assessment and visualisation process in the following code chunks, starting with constructing the model with only the 8 statistically significant variables.
Building Global Model
model_refined <- glm(status ~ distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_clean,
family = binomial(link = "logit"))
blr_regress(model_refined) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4746 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3540 0.1055 3.3541 8e-04
distance_to_tertiary_road 1 1e-04 0.0000 4.9096 0.0000
distance_to_city 1 0.0000 0.0000 -5.2022 0.0000
distance_to_town 1 0.0000 0.0000 -5.4660 0.0000
is_urbanTRUE 1 -0.2667 0.0747 -3.5690 4e-04
usage_capacity1000 1 -0.6206 0.0697 -8.9081 0.0000
water_source_cleanProtected Shallow Well 1 0.4947 0.0850 5.8228 0.0000
water_source_cleanProtected Spring 1 1.2790 0.4384 2.9174 0.0035
water_point_population 1 -5e-04 0.0000 -11.3902 0.0000
local_population_1km 1 3e-04 0.0000 19.4069 0.0000
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7349 Somers' D 0.4697
% Discordant 0.2651 Gamma 0.4697
% Tied 0.0000 Tau-a 0.2320
Pairs 5585188 c 0.7349
---------------------------------------------------------------We check and see that the remaining variables are all statistically significant to the linear regression model (p-values < 0.05).
Confusion Matrix for Global Model
The code chunk below calculates and displays the confusion matrix for the refined model. We will discuss the results together with that for the refined gwLR model in the subsequent subsection.
blr_confusion_matrix(model_refined, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1300 743
1 814 1899
Accuracy : 0.6726
No Information Rate : 0.4445
Kappa : 0.3348
McNemars's Test P-Value : 0.0761
Sensitivity : 0.7188
Specificity : 0.6149
Pos Pred Value : 0.7000
Neg Pred Value : 0.6363
Prevalence : 0.5555
Detection Rate : 0.3993
Detection Prevalence : 0.5704
Balanced Accuracy : 0.6669
Precision : 0.7000
Recall : 0.7188
'Positive' Class : 1Determining Fixed Bandwidth for GWR Model
bw.fixed_refined <- bw.ggwr(status ~ distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_sp,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE, # for fixed bandwidth
longlat = FALSE) # input data have been converted to projected CRSbw.fixed_refined[1] 2377.371
The output for bw.fixed_refined is given above. We will use this optimal fixed distance value for model assessment in the next subsection.
Model Assessment
gwlr.fixed_refined <- ggwr.basic(status ~ distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = wp_sp,
bw = 2377.371,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413 gwlr.fixed_refined ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 16:40:56
Call:
ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city +
distance_to_town + is_urban + usage_capacity + water_source_clean +
water_point_population + local_population_1km, data = wp_sp,
bw = 2377.371, family = "binomial", kernel = "gaussian",
adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-129.368 -1.750 1.074 1.742 34.126
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.540e-01 1.055e-01 3.354 0.000796
distance_to_tertiary_road 1.001e-04 2.040e-05 4.910 9.13e-07
distance_to_city -1.764e-05 3.391e-06 -5.202 1.97e-07
distance_to_town -1.544e-05 2.825e-06 -5.466 4.60e-08
is_urbanTRUE -2.667e-01 7.474e-02 -3.569 0.000358
usage_capacity1000 -6.206e-01 6.966e-02 -8.908 < 2e-16
water_source_cleanProtected Shallow Well 4.947e-01 8.496e-02 5.823 5.79e-09
water_source_cleanProtected Spring 1.279e+00 4.384e-01 2.917 0.003530
water_point_population -5.098e-04 4.476e-05 -11.390 < 2e-16
local_population_1km 3.452e-04 1.779e-05 19.407 < 2e-16
Intercept ***
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
is_urbanTRUE ***
usage_capacity1000 ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
water_point_population ***
local_population_1km ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.9 on 4746 degrees of freedom
AIC: 5708.9
Number of Fisher Scoring iterations: 5
AICc: 5708.923
Pseudo R-square value: 0.129406
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2377.371
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -3.7021e+02 -4.3797e+00 3.5590e+00
distance_to_tertiary_road -3.1622e-02 -4.5462e-04 9.1291e-05
distance_to_city -5.4555e-02 -6.5623e-04 -1.3507e-04
distance_to_town -8.6549e-03 -5.2754e-04 -1.6785e-04
is_urbanTRUE -7.3554e+02 -3.4675e+00 -1.6596e+00
usage_capacity1000 -5.5889e+01 -1.0347e+00 -4.1960e-01
water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01 6.2378e-01
water_source_cleanProtected.Spring -1.3630e+03 -5.3436e+00 2.7714e+00
water_point_population -2.9696e-02 -2.2705e-03 -1.2277e-03
local_population_1km -7.7730e-02 4.4281e-04 1.0548e-03
3rd Qu. Max.
Intercept 1.3755e+01 2171.6375
distance_to_tertiary_road 6.3011e-04 0.0237
distance_to_city 1.5921e-04 0.0162
distance_to_town 2.4490e-04 0.0179
is_urbanTRUE 1.0554e+00 995.1841
usage_capacity1000 3.9113e-01 9.2449
water_source_cleanProtected.Shallow.Well 1.9564e+00 66.8914
water_source_cleanProtected.Spring 7.0805e+00 208.3749
water_point_population 4.5879e-04 0.0765
local_population_1km 1.8479e-03 0.0333
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2815.659
AIC : 4418.776
AICc : 4744.213
Pseudo R-square value: 0.5691072
***********************************************************************
Program stops at: 2022-12-17 16:41:23 The AIC values of the 4 models are summarised in the table below.
| AIC values | Global model | gwLR |
|---|---|---|
| All variables | 5,712 | 4,414.606 |
| Statistically significant variables only | 5,708.9 | 4,418.776 |
We see that both gwLR models have lower AIC values than their global model counter parts. Between the 2 gwLR models, the one with all 10 variables have a slightly lower AIC value (4,414.606) than the one with only the 8 statistically significant variables (4,418.776).
Converting SDF into sf Data Frame
gwr.fixed_refined <- as.data.frame(gwlr.fixed_refined$SDF)Assigning Cutoff Value of 0.5
gwr.fixed_refined <- gwr.fixed_refined %>%
mutate(most = ifelse(
gwr.fixed_refined$yhat >= 0.5, T, F))Confusion Matrix for gwLR
We similarly call the confusion matrix and statistics using confusionMatrix() of caret in the code chunk below.
gwr.fixed_refined$y <- as.factor(gwr.fixed_refined$y)
gwr.fixed_refined$most <- as.factor(gwr.fixed_refined$most)
CM_refined <- confusionMatrix(data = gwr.fixed_refined$most,
reference = gwr.fixed_refined$y,
positive = "TRUE")
CM_refinedConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1833 268
TRUE 281 2374
Accuracy : 0.8846
95% CI : (0.8751, 0.8935)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7661
Mcnemar's Test P-Value : 0.6085
Sensitivity : 0.8986
Specificity : 0.8671
Pos Pred Value : 0.8942
Neg Pred Value : 0.8724
Prevalence : 0.5555
Detection Rate : 0.4992
Detection Prevalence : 0.5582
Balanced Accuracy : 0.8828
'Positive' Class : TRUE
We see that the accuracy (0.8837 vs 0.6739), sensitivity (0.8628 vs 0.7207) and specificity (0.9005 vs 0.6154) values have all improved from the non-gwLR global model. By using the gwLR model, we can explain the non-functional water points better now which allows better management of water points through localised strategies (e.g. look at the local neighbourhood regions within Osun state).
The performance measures of the 4 logistic regression models are summarised in the table below.
| Performance Measure | Global regression with 10 variables | gwLR with 10 variables | Global regression with 8 variables | gwLR with 8 variables |
|---|---|---|---|---|
| Accuracy | 0.6739 | 0.8837 | 0.6726 | 0.8846 |
| Sensitivity | 0.7207 | 0.9005 | 0.7188 | 0.8986 |
| Specificity | 0.6154 | 0.8628 | 0.6149 | 0.8671 |
We see that the model accuracy and specificity improve very slightly by removing the non-statistically significant variables from the gwLR model, but the sensitivity drops slightly. Nevertheless, as we would be more interested in finding non-functional water points for maintenance etc., the gwLR model with 8 variables would be more useful with a higher specificity.
Visualising gwLR
Now we combined the prediction from the refined gwLR model with the water point sf data frame with selected variables for visualisation.
gwr_sf.fixed_refined <- cbind(wp_sf_select, gwr.fixed_refined)We can similarly visualise the actual versus predicted functional (more red) and non-functional (more yellow) water points using the code chunk below.
tmap_mode("view")
actual <- tm_shape(osun) +
tmap_options(check.and.fix = TRUE) +
tm_polygons(alpha = 0.4) +
tm_shape(wp) +
tm_dots(col = "status",
alpha = 0.6,
palette = "YlOrRd") +
tm_view(set.zoom.limits = c(9, 12))
prob_T_refined <- tm_shape(osun) +
tm_polygons(alpha = 0.4) +
tm_shape(gwr_sf.fixed_refined) +
tm_dots(col = "yhat",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(9, 12))
tmap_arrange(actual, prob_T_refined,
asp = 1, ncol = 2, sync = TRUE)We see that the predictions are largely aligned with the actual status of the water points, in line with the 88% accuracy rate.